Mathematical Sciences: Weighted Norm Inequalities and Partial Differential Equations

该项目涉及数学分析的两个子领域的问题：偏微分方程和与高斯测度相关的调和分析。 该项目有两个目标。 第一个是分析一类具有非光滑系数的散度形式的简并抛物型方程的解的行为。 这些系数可以为零、无穷大或两者兼而有之。 已知这些方程可模拟非均质和非各向同性材料中的温度扩散情况。 这项工作要考虑的基本问题包括寻找暗示解规律性的条件、确定哈纳克原理对于非负解是否有效以及理解方程基本解的行为。 本研究将使用庞加莱和索博列夫类型的加权范数不等式。 第二个目标涉及由 Ornstein-Uhlenbeck 半群生成的一类变换在高斯测度可积函数空间中的行为。 在高斯背景下，这些变换扮演着 M. Riesz 的经典奇异积分的角色。 特别令人感兴趣的是建立弱类型估计，其常数与维度无关。 本研究将集中于 Calderon-Zygmund 理论中最大函数的使用，但该函数并不继承经典最大函数的所有属性。 然而，有证据表明它对于高斯测度来说仍然是弱类型。 这是首先要解决的问题。

This project is concerned with problems in two subareas of mathematical analysis: partial differential equations and harmonic analysis related to the Gaussian measure. The project has two objectives. The first is to analyze the behavior of solutions of a class of degenerate parabolic equations in divergence form with non-smooth coefficients. Such coefficients may be zero, infinite or both. The equations are known to model cases of diffusion of temperature in a non-homogeneous and non-isotropic material. Basic questions to be considered in this work include finding conditions which imply regularity of solutions, determining whether a Harnack principle is valid for non-negative solutions and understanding the behavior of the fundamental solutions of the equations. Weighted norm inequalities of the Poincare and Sobolev type will be used in this study. The second objective concerns the behavior, in the space of functions integrable with respect to Gaussian measure, of a class of transforms generated by the Ornstein-Uhlenbeck semigroup. These transforms play the role, in the Gaussian context, of the classical singular integrals of M. Riesz. Of particular interest will be the establishment of weak-type estimates with constants bounded independently of dimension. This investigation will center on the use of a maximal function as in the Calderon- Zygmund theory, but this function does not inherit all the properties of the classical maximal functions. However, there is evidence to suggest that it is still of weak type with respect to Gaussian measure. This is the first problem to be resolved.